$f_n$ is differentiable on $[a,b]$ and $\lim_{n\to\infty}f_n(x)$ exists for each $x$, Suppose $|f_n'(x)|\leq M$, can we show $f$ is differentiable?
$f_n$ is differentiable on $[a,b]$ and $\lim_{n\to\infty}f_n(x)$ exists for each $x$, Suppose $|f_n'(x)|\leq M$, can we show $f$ is differentiable?
Clearly, $f_n$ is uniformly continuous. but we have not the uniform continuity of $f_n'$. So I think the above statement is wrong, but I could not find such a counterexample. Help.
A counterexample is $$ f_n(x) = \sqrt{\frac 1n + x^2} \, . $$ The $f_n$ are differentiable on $\Bbb R$ with $|f_n'(x)| \le 1$ and converge uniformly to $|x|$.